Compared with gold and silver, cheap copper has attracted more attention and can potentially be applied in non-enzymatic electrochemical sensors due to its excellent conductivity and catalytic activity. In this paper, copper nanoplates were rapidly synthesized using copper bromide as the copper precursor, polyethyleneimine as the stabilizer, and ascorbic acid as a reducing agent in the presence of silver nanoparticles at a reaction temperature of 90 °C. The Cu nanoplates with an average side length of 10.97 ± 3.45 μm were obtained after a short reaction time of 2 h, demonstrating the promoting effect of an appropriate amount of silver nanoparticle on the synthesis of Cu nanoplates. Then, the electrochemical dopamine sensor was constructed by modifying a glass carbon electrode (GCE) with the Cu nanoplates. The results obtained from the test of cyclic voltammetry and chronoamperometry indicated that the Cu-GCE showed a significant electrochemical response for the measurement of dopamine. The oxidation peak current increased linearly with the concentration of dopamine in the range of 200 µmol/L to 2.21 mmol/L, and the corresponding detection limit was calculated to be 62.4 μmol/L (S/N = 3). Furthermore, the anti-interference test showed that the dopamine sensor was not affected by a high concentration of ascorbic acid, glucose, uric acid, etc. Therefore, the constructed Cu-GCE with good selectivity, sensitivity, and stability possesses a high application value in the detection of dopamine.